The results of \(\Delta N_{\rm eff}=N_{\rm eff}-N^{\rm SM}_{\rm eff}\)  as a function of cosmic time \(t\)  are shown in Fig. REF , where \(N^{\rm SM}_{\rm eff}=3.045\)  is considered [1]}, [2]}, [3]}.
Here we fix \(m_{\chi }=10\)  GeV and \(m_{\phi }=50\)  GeV for illustration. For \(f_\phi =1\) , a lifetime \(\tau _{\phi }=10^{10}\)  s has been excluded by Planck measurement. Meanwhile, a lifetime \(\tau _{\phi }=10^{9}\)  s with \(f_\phi =1\)  leads to \(\Delta N_{\rm eff}=0.12\) , which is within the reach of future CMB S4 experiment. And a lifetime down to about \(10^8\)  s will beyond the scope of future limit. If fixing \(\tau _{\phi }=10^9\)  s, it is clear that \(f_\phi =10\)  predicts a too large value of \(\Delta N_{\rm eff}\) . To avoid future CMB S4 limit, we need \(f_\phi \sim \mathcal {O}(0.1)\)  with \(\tau _{\phi }=10^9\)  s.
Approximately, constraints from \(N_{\rm eff}\)  are proportional to the product of \(\tau _{\phi }\)  and \(f_{\phi }\) . Currently, the Planck result requires \(f_{\phi }^2\tau _{\phi }\lesssim 5\times 10^9\)  s, and the future CMB S4 could push to \(f_{\phi }^2\tau _{\phi }\lesssim 2.5\times 10^8\)  s. The corresponding exclusion limits are also shown in Fig. REF .
